Width 0 .45 m

2.2 legged robots and the mechanical structure of robotic legs 13

Table 2.1: Specification of ASIMO

Mass 52 kg

Height 1 .2 m

Width 0 .45 m

Depth 0 .44 m

Moving velocity 0 -1.6 km/h

Biped cycle variable cycle / step

Grasping force 0 .5 kg

Actuator servo + harmonic drive Leg force sensor 6 -D force/torque Body sensor Gyro + acceleration Power supply 38 V/10Ah

Head 2

Shoulder 2

DoF

(s) per arm

Head 3

Elbow 1

Wrist 1

Finger 1

Hip 3

DoF

(s) per leg

Knee 1

Ankle 2

TITAN

IV), electrical linear actuators with coupled drive (

VII, hydraulic lin- ear actuator (

XI) and pulley-wire drive (

VIII,

XIII [Kitano et al., 2013 ]) .

2.2.3 MIT—Cheetah

MIT Cheetah (shown in Fig. 2 .9) is a quadruped robot developed by Sangbae Kim’s

team at MIT since around 2012 [McKenzie, 2012 ]. This robot was designed to-

wards high-speed locomotion and low cost-of-transport (

) featured with novel

actuation systems and innovative mechanical structure. To realize fast locomo-

tion and high efficiency of locomotion, several specific principles are proposed

and implemented [Seok et al., 2015 ]. MIT Cheetah is equipped with eight large-

diameter brushless-direct-current motors (

) that will take the advantage of

high torque-mass ratio. Every two motors are arranged co-axially forming a drive

unit mounted at the shoulder/hip joints of four legs. The shoulder/hip joint of

each leg is powered by one

motor directly and the elbow/knee joint of every

leg is driven by the other motor placed at the shoulder/hip through a four-bar

14 b a c k g r o u n d a n d r e l at e d w o r k

Figure 2.7: Kinematic configuration of a humanoid robot leg. J0− J₂are theDoFsbelonging to the hip joint; J3is the knee joint; J4, J5belongs to the ankle joint. The figure was adopted from [Kajita et al.,2014].

steel linkage. This leg configuration can make the

near the shoulder/hip joint and, therefore, the inertia of the whole leg during the swing motion will be re- duced. A custom-made, single-stage low gear-ratio (5.8 : 1) planetary gear box is used with motors to increase the output torque meanwhile keeping the introduced friction and impedance at a low level. Because of the low mechanical impedance of the leg, the force transmission is almost ’transparent’ from actuator to end ef- fector. Thus the external force applied on foot can be measured directly by joint torque— the motor current with a high bandwidth [Seok et al., 2012 ]. Besides the shoulder/hip joint and elbow/knee joint of each leg, a passive joint at the distal end near the foot is designed to link the foot to the elbow/knee joint by a tendon made of Kevlar, which can release the impact load applied on the structural com- ponent of the leg while landing. MIT Cheetah’s structure is made of polyurethane foam and resin that features high strength and low density, the utilization of ten- dons enables the structural parts to carry the compression load and the tendons to afford the extension load principally [Ananthanarayanan et al., 2012 ]. The legs of MIT Cheetah are designed and implemented based on a bioinspired approach;

the front and rear legs adopt distinct dimensions to acquire optimized torque and

velocity profiles respectively, on the contrary most quadruped robots tend to use

identical structural design for all four legs. The MIT Cheetah has a flexible spine

between front and rear legs. During high-speed running, the spine of the MIT

2.2 legged robots and the mechanical structure of robotic legs 15

(a) (b)

(c) (d)

Figure 2.8:TITANseries robots (a)TITANIII (b)TITANIV.(c)TITANVII (d)TITANXI.

Cheetah can be actuated and arched by the rear legs through in-phase movement mode then the stride lengths of rear legs can be increased.

MIT Cheetah is powered by a on-board Li-Po battery. The electronics driving motors are designed for energy regeneration. When the robotic leg is applied a load to break the robot, the motor drive functions as a boost converter that can convert the negative work made by the motor to a higher voltage than that of the battery and then recharge the battery. According to published experimental result, MIT Cheetah can run at a speed up to six m/s on the experimental setup com- posed of a treadmill and a boom constraining robot within a plane. Meanwhile, the total

, which includes both the mechanical energy loss and heat loss by ac- tuator, is about 0.5 [Seok et al., 2015 ]. The specifications of MIT Cheetah are listed in Table 2 .2 [McKenzie, 2012 ; Seok et al., 2015 ].

2.2.4 BigDog

BigDog is a hydraulic quadruped robot built at Boston Dynamics, which is a spin-

off from MIT founded by Marc Raibert et al. in 1992. BigDog was developed under

funding by the Defense Advanced Research Projects Agency (

) , targeting

16 b a c k g r o u n d a n d r e l at e d w o r k

Figure 2.9: MIT Cheetah quadruped robot.

rough-terrain mobility superior to existing wheeled and tracked vehicles [Raibert et al., 2008 ]. The primitive prototype of BigDog was first released in 2005 (referred to as BigDog 2005 in this dissertation) and was designed to be relevant to hu- man size in terms of mobility, speed, and load carrying ability. BigDog 2005 is about one meter tall, one meter long, and 0.3 meter wide and weights 90 kg. A 17 -horsepower combustion engine is adopted as onboard power supply to drive a variable displacement hydraulic pump, providing 3000

(equivalent to 20.7 Mpa) pressure for the hydraulic system. According to [Buehler et al., 2005 ], each of the four legs has four

in serial: one passive linear pneumatic compliance in the lower leg connected with the foot, one powered knee joint, and two pow- ered hip joints, as shown in Fig. 2 .10a. All 12 active joints are driven by identical servo actuators— a custom hydraulic cylinder integrated a with servo valve, linear positional sensor, and load cell. As Boston Dynamics seldom gives details about their robots, it is speculated, according to released figures and videos, that all four legs have the same mechanical structure and, at each joint, a typical crank-slider mechanism is designed, in which the base and crank are structural links of leg, re- spectively. Moreover, for BigDog 2005, the knee joints of all four legs bend toward one direction, the benefit of this configuration may be the simplicity of control.

BigDog 2005 can move using diverse gaits such as trotting and walking. In exper- iments, the robot can walk up and down 35 degree inclines, trot at speeds up to 0 .8m/s and carry more that 50 kg of payload.

In 2006, an improved version of BigDog (referred to as BigDog 2006) was pre-

sented by released videos. According to related videos and little published infor-

mation, BigDog 2006 has analogous specifications and mechanical structure to the

previous BigDog 2005 but all its knee joints bend inwards: front and rear legs in

opposite directions. BigDog 2006 is shown in Fig. 2 .10b.

2.2 legged robots and the mechanical structure of robotic legs 17

Table 2.2: Specification of MIT Cheetah

Mass 33 kg

Front leg mass 3 .25 kg

Rear leg mass 3 .6 kg

Torso length 0 .65 m

Front leg length 0 .24 - 0.48 m

Rear leg length 0 .23 - 0.5 m

Front leg shoulder motion range 150

Front leg elbow motion range 107

Rear leg hip motion range 150

Rear leg knee motion range 85

The latest version of the BigDog robot was released through online videos in 2008 (referred to as BigDog 2008), as shown in Fig. 2 .10c. Several significant spec- ifications and parameters of BigDog 2008 were introduced in [Raibert et al., 2008 ] and summarized in Table 2 .3 but details of mechanical design were still not re- leased. The mechanical system of BigDog 2008 and the precedent are alike: four legs are mounted at the corners of the trunk with opposite orientation. However, BigDog 2008 has one more powered joint in each leg; thus, the leg of BigDog 2008 is a kinematically redundant mechanism in leg plane, as shown in Fig. 2 .11a. It is speculated that this design may have advantages in optimizing the load dis- tribution between joints in one leg. The robot is powered by a two-stroke single cylinder engine with around 17 hp output and all active joints are driven by a custom actuator package integrated with a high-bandwidth (greater than 250 Hz) servo valve provided by Moog, a low-friction hydraulic cylinder, a position sensor, and a force sensor (Fig. 2 .11b and Fig. 2 .11c) [Boston Dynamics, 2008 ]. Crank-slider mechanisms are used in joints as well. Combined with the length of the cylinder, the force of cylinder, and the kinematics of joint mechanism, the joint position and force can be computed. As a result, joint position and torque can be controlled ac- tively. In released videos, BigDog presented superior capability of dynamic balanc- ing and traversability through tough terrains such as rocky slopes, sandy beaches and mountainous district to the legged robots developed ever before.

BigDog inspired a trend of research on the quadruped robots actuated hydrauli- cally and featured with highly dynamic motions. The leg structure of BigDog was also adopted and used by many quadruped robots developed subsequently such as [Li et al., 2013 ; Cho et al., 2013 ].

2.2.5 HyQ and HyQ2Max

HyQ

is a hydraulically actuated quadruped robot developed by the Dynamic Legged

Systems (DLS) Lab at the Istituto Italiano di Tecnologia (IIT) [Semini, 2010 ; Semini

18 b a c k g r o u n d a n d r e l at e d w o r k

Table 2.3: System Overview of BigDog 2008

Dimensions 1 .1 × 0.3 × 1 m, L× W× H

DoFs

per leg 4 active + 1 passive

Mass 109 kg

Installed power 17 (11) hp (kW)

Payload capability 50 , nominal kg

154 , max. kg

Hydraulic pressure 3000 (20.7) psi (Mpa)

Endurance 10 (2.5h hike) km

Speed 0 .2, crawl m/s

1 .6, walking trot m/s 2 , flying trot m/s 3 .1, bound in lab m/s

(a) (b) (c)

Figure 2.10: Picture selection of various versions of BigDog: (a) BigDog 2005; (b) BigDog 2006and (c) BigDog 2008.

et al., 2011 ], as shown in Fig. 2 .12a. Hydraulically actuated Quadruped (

) has been designed to perform highly-dynamic motions such as trotting and jumping aimed at traveling over rough terrain in the natural environment. Diverse potential applications are targeted by this robot such as search and rescue, forestry technol- ogy, and construction. It roughly has the dimensions of a goat, i.e. 1.0 m×0.5 m×0.98 m.

The leg length ranges from 0.34 - 0.79 m and the hip-to-hip width is 0.75 m.

’s weight is approximately 80 kg; it slightly varies depending on the exteroceptive sensors, such as cameras and laser scanner. The robot is equipped with 12 active

DoFs

without passive joints. Three joints in each leg,

,

and

, as shown

in Fig. 2 .12b, are arranged in serial from trunk to foot and the knee joints of all

four legs bend inwards. We define this knee arrangement as the

configuration,

as shown in Fig. 2 .12c. The

robot is actuated by eight hydraulic cylinders

(

and

) and four hydraulic rotary motors (

), which are all driven by

high-bandwidth (greater than 250 Hz) servo valves of Moog. At every piston rod

2.2 legged robots and the mechanical structure of robotic legs 19

(a) (b)

(c)

Figure 2.11: Structure and key components: (a) Structure of BigDog 2008; (b) Engine and (c) Hydraulic actuator package.

end, there are load-cells connected that measure the forces of the pistons and, con- sequently the joint torques can be obtained combined with the mechanism kine- matics. Similarly, a custom torque sensor provides direct measurement of the Hip Abduction/Adduction (

) torques. High-resolution encoders, both relative and absolute encoders, are mounted along with the joint axes, then joint positions are able to be measured directly. Owing to no passive joint, all joints are fully torque- controlled which enables to actively control the compliance of legs[Boaventura et al., 2015 ; Barasuol et al., 2013 ]. The controller of

is a onboard computers running real-time Linux. The computer processes the low-level control (hydraulic- actuator control) at 1 KHz which communicates with the proprioceptive sensors through EtherCAT boards. The specification of the

robot is summarized in Table 2 .4.

HyQ2Max is an evolutionary version of

robot published in [Semini et al.,

2017 ] in 2016, shown in Fig. 2 .13a. HyQ2Max has a similar specification to

’s

(listed in Table 2 .4), additionally HyQ2Max owns sturdier mechanical structure

20 b a c k g r o u n d a n d r e l at e d w o r k

(a)

- +

+ - +

Y Z X

(b)

KFE HFE HAA

x z

y

(c)

Figure 2.12:HyQand its kinematic configuration. (a) A photograph ofHyQ. (b) Leg struc- ture of HyQ. (c) Kinematic structure of theHyQrobot, adopted from [Semini, 2010; Semini et al., 2017]. The robot has four legs with identical mechani- cal structure, and mounted oppositely. Legs are referred to as Left-Front (LF), Right-Front (RF), Left-Hind (LH) and Right-Hind (RH) legs respectively. Each leg has three joints Hip Abduction/Adduction (HAA), Hip Flexion/Extension (HFE) and Knee Flexion/Extension (KFE). Note that we useHyQ’s naming rule for legs, joints and coordinate frame definition all over this dissertation.

and more robust joint actuation prepared for natural environment deployment.

HyQ2Max is actuated by 8 hydraulic rotary motors placed at

and Hip Flex-

ion/Extension (

) joints and four hydraulic cylinders in Knee Flexion/Exten-

2.2 legged robots and the mechanical structure of robotic legs 21

sion (

) joints. The joint configuration of HyQ2Max can be seen in Fig. 2 .13b.

HyQ2Max takes the

knee configuration in use as well as

. Differently, the leg plane formed by the upper leg and lower leg of HyQ2Max is distant from the axis of

joint with a 0.1-meter sideways offset. This design enables the

HFE

joint to swing within a extender range comparing to

, up to 270 degrees.

Similarly by the using of multi-bar mechanism, the motion ranges of knee joints are also enlarged considerably, which can be up to 160 degrees. Consequently, thanks to the large joint motion range, HyQ2Max is able to achieve more versatile motions such as self-righting and turnover and leg folding. The sensing system of HyQ2Max is the same as

’s. Position sensors are mounted co-axially with each joint: for

and

encoders directly measure the angular positions of the hydraulic motor shafts; for

encoders is arranged at the knee joint to measure the angular position in joint space rather than in actuator space like BigDog. Cus- tom torque sensors are equipped with

and

joints to feedback the output torque directly. A loadcell connected with the cylinder body is utilized to measure the force exerted by hydraulic cylinder and together with the force transmission ratio of multi-bar linkage in knee joint to compute the joint torque of

. The specification of HyQ2Max is listed in Table 2 .4.

(a) (b)

Figure 2.13: HyQ2Max and its joint configuration. (a) A photograph of HyQ2Max. (b) Joint configuration of the leg of the HyQ2Max robot. Similar to HyQ, each leg of HyQ2Max has three joints: Hip Abduction/Adduction (HAA), Hip Flex- ion/Extension (HFE) and Knee Flexion/Extension (KFE). BetweenHAAandHFE, there is a 0.1-meter sideways offset. HAAand HFEjoints are actuated by hy- draulic motor that can output rotational motions directly;KFEis actuated by a hydraulic cylinder combining with a four-bar linkage as transmission. The electronics ofKFEis integrated into upper leg.

22 b a c k g r o u n d a n d r e l at e d w o r k

Table 2.4: System Overview of HyQ and HyQ2Max

HyQ HyQ2Max

Dimensions 1 .0×0.5×0.98 1 .3×0.5×0.92 m, L×W×H

Mass 80 kg, (off-board power)

Hip-hip distance 0 .747 0 .887 m, fore-aft

0 .414 0 .194 m, left-right

Link length & mass 0 .08 / 2.9 0 .10 / 3.54 m / kg, hip 0 .35 / 2.6 0 .36 / 4.95 m / kg, upper leg 0 .36 / 0.8 0 .38 /1.40 m / kg, lower leg

Active

12 12

HAA actuator double-vane motor

HFE actuator cylinder single-vane motor

KFE actuator asymmetric cylinder

Joint motion range 90 –120–120 90 –270–166

, HAA–HFE–KFE

Max. torque, HAA 120 120

Nm, at 20 Mpa Max. torque, HFE 181 , max. 245

Max. torque, KFE 181 , max. 250 , max

Position sensors 80 , 000 ppr 18 bit absolute encoder

Torque sensor/Loadcell T-L-L T-T-L HAA–HFE–KFE

Onboard computer Pentium i5 with real-time Linux

Controller rate 1 kHz, (EtherCAT)

2.3 summary 23

2.3 Summary

According to the introduction and analysis above, some conclusions can be drawn as below. These conclusions will be the postulates of the research in subsequent chapters.

• The legs of mammals like horse primary perform motions in a plane paral- leling with the saggital plane of horse, although animals and its legs can do 3 D motions.

• The typical and basic configuration of quadruped robot consists of one trunk and four identical legs. The hip joint of each leg are arranged at the four corners of the trunk forming a rectangle. Legs can be mounted with the same orientation or opposite orientations.

• A two-

mechanism, one for extension/flexion like knee joint; the other one mainly for the direction changing like hip, is the basic unit of leg, which is in common in the robots and legged locomotion models presented above.

Additional

could be considered as interfaces of this basic unit connect- ing with the body of robot and foot.

• Based on the models of legged locomotion, when running, the

and hip joint will have obvious vertical displacement; in contrast, for low speed mo- tion such as walking, the hip and

are able to remain nearly constant height.

• Existing devices/facilities are lack of the functions of providing complex and varying terrains for legged robot performance test and evaluation.

• Comparing with other types of legged robots, e.g., humanoid robots, electric-

powered robots, etc, hydraulically actuated quadruped robots have better

performance in the aspects of speed, payload, and terrain adaption.

Study on the Morphological Parameters 3

Considering Ditch Crossing Traversability

At the beginning of designing a quadruped robot targeting at rough terrain mo- bility, the first question faced may be how to determine the morphological param- eters such as trunk size and limb length for the robot to meet the requirement of overcoming obstacles in given environment. We suppose the capability of over- coming obstacle are highly correlated with the morphological parameter selection of a quadruped robot. For a specified quadruped robot, its obstacle traversabil- ity should have a limit determined by morphological parameters which is inde- pendent from the other factors such as control, actuation. So it is necessary to explore the traversability limit caused by morphological parameters, otherwise improper morphological parameters may become a principal limitation for the mobility of quadruped robot. This chapter presents the influence of morpholog- ical parameters of quadruped robots on the capability of crossing a ditch, since ditch is one of the most typical obstacle and able to be quantified by the width of ditch simply. In the first section, the basic model and morphological parame- ters of quadruped robots are introduced and defined. Then movement sequences of quadruped robot during ditch crossing are derived. Subsequently a number of simulations are performed with varying morphological parameters to explore the effect on ditch crossing capability. The results are presented and analyzed conse- quently. The main content in this chapters has been published in [Gao et al., 2016 ].

3.1 Basic Model and Morphological Parameters

In this section we present the model describing generalized quadruped robots with mammal configuration, define morphological parameters for quadruped robots and constraint conditions considered in the research.

3.1.1 Basic Model and Morphological Parameters

In order to study the generalized effect of morphological parameters on ditch crossing capability, we define a basic model to represent quadruped robots with mammal configuration. Comparing with sprawling configuration, the first joint of mammal configuration rotates about the roll axis of the robot and its feet usually locates beneath the trunk [Kitano et al., 2013 ]. This basic model contains minimum parts and joints for comprising a quadruped robot with four identical articulated

25

26 s t u d y o n t h e m o r p h o l o g i c a l pa r a m e t e r s c o n s i d e r i n g d i t c h c r o s s i n g t r av e r s a b i l i t y

Figure 3.1: Basic model and morphological parameters of quadruped robots with mam- mal configuration. The quadruped robots with mammal configuration can be abstracted as a 9-part rigid body system, including one trunk and four iden- tical two-link legs. Three types of parameters are selected as morphological parameters, including joint-to-joint (or distal end) distance, mass of each part and theCoM position of each part. Two Cartesian coordinate frames are set at point O, the geometric center of rectangle formed by four hips in trunk and move with the robot. Ox0z₀ is world coordinate frame, which maintains un- changed orientation, keeping its x0 axis and z0 axis aligned with horizontal and vertical direction respectively. Oxrzris the robot coordinate frame, aligns its x+ axis with the heading direction of robot. The angle from x0+to xr+is defined as pitch angle θ of the trunk.

legs. Figure 3 .1 presents the basic model, and in this chapter, we use the convention in [Semini, 2010 ] to name corresponding joints and components of robot. The basic model consists of a one-piece trunk or torso and four identical legs attached at each corner of trunk. Each leg has two rotational joints with their axes parallel indicating hip joint (

) linking upper leg to trunk and knee joint (

) between upper leg and lower leg. Consequently there are 8

in joints and 3

of the whole robot, which are linear Degree of Freedom (

) along x, z and the pitch of trunk θ, in sagittal plane.

At the beginning of robot design, the trunk length and mass are usually first- determined parameters according to specific application requirement e.g. desired payload and its size. Then other morphological parameters could be decided based

1 The sagittal plane is spanned by gravity vector and the robot’s heading direction.

3.1 basic model and morphological parameters 27

Table 3.1: Parameters and Variables

Definition Normalized Parameters Variables

Length of trunk (front/hind hip distance) 1 /

Length of upper leg ¯l

/

Length of lower leg ¯l

/

Mass of trunk 1 /

Mass of upper leg m ¯

/

Mass of lower leg m ¯

/

Trunk

position in robot frame r ¯

/

Upper leg

position with respect to hip r ¯

/

Lower leg

position with respect to knee r ¯

/

Positional vector of hips ¯h

/

Pitch attitude / θ

Hip joint angle / α

Knee joint angle / β

on the parameters of trunk. Additionally, in this thesis, to investigate and compare the effect on quadruped robots with various size, morphological parameters are normalized with respect to the trunk length and denoted with a bar superscript.

The distance between front and hind hips at one side is defined as 1 unit length and the mass of trunk is defined as 1 unit mass as well. Then other morpholog- ical parameters are normalized as dimensionless ratios relative to trunk’s param- eters accordingly. Furthermore based on our previous experience from

and HyQ2Max, upper and lower legs of quadruped robot could be regarded as bar-like parts, whose

lie in the axis of bar, as a result positions of

of upper and lower leg can be expressed as ratios ¯r

⊂ [0, 1] with respect to the total length of corresponding link.

position of trunk is able to vary in sagittal plane of robot, its location is noted as ¯ r

. The angular variables of joints α

, β

and the index of legs i = 1 ∼ 4 can be seen in Fig. 3 .1 as well. Definition of parameters and vari- ables are listed in Table 3 .1. Thus according to the definition of parameters above, normalized

position of the whole robot expressed in world coordinate frame

¯

p can be computed as Eq. ( 2 ).

¯

p = R(θ) r ¯

+ m ¯

P

i=1

¯h

+ R

(α

) ¯ l

r ¯

+ ¯ m

P

i=1

¯h

+ R

(α

) ¯ l

+ R

(β

) ¯ l

r ¯

1 + 4 ( m ¯

+ m ¯

)

(2)

where R

(α

) , R

(β

) are the rotation matrices of hip, knee with respect to cor-

responding joint axes and R(θ) is the rotation matrix from robot frame to world

frame. Since we focus on the effects induced by morphological parameters, footholds

are specified according to geometric and critical postures during ditch crossing in

which the

of the whole robot just locates at the edge of the ditch, then joint

28 s t u d y o n t h e m o r p h o l o g i c a l pa r a m e t e r s c o n s i d e r i n g d i t c h c r o s s i n g t r av e r s a b i l i t y

angles α

, β

are figured out based on inverse kinematics and knee configuration.

Knowing foothold locations and

position of robot from Eq. ( 2 ) at the same time, longitude stability can be judged. If the horizontal projection of

is lo- cated inside support polygon formed by supporting feet, we conclude this posture is achievable (i. e. stacially stable).

3.1.2 Knee Configuration

For most of quadruped robots, legs have identical mechanism and the knee joints are designed to extend/flex in single side of leg so that multiple solutions of in- verse kinematics can be avoid. Thus considering different knee bending direction, there will be four types configurations available, shown in Fig. 3 .2. In actual prac- tice all four configurations have been engineered by distinct robots already, but the influence and characteristics underlying have rarely been studied (e.g. [Witte et al., 2001 ; Xiuli et al., 2005 ]). Distinct knee configurations not only induce the difference in workspace of feet but also lead to variation in mass distribution and affect the position of

and stability margin.

−0.5 0

0.5 −0.5 0

0.5

−0.4

−0.2 0 0.2

y x

z

−0.5 0

0.5 −0.5 0

0.5

−0.4

−0.2 0 0.2

x y

z

−0.5 0

0.5 −0.5 0

−0.4 0.5

−0.2 0 0.2

y x

z

−0.5 0

0.5 −0.5 0

0.5

−0.4

−0.2 0 0.2

y x

z

Figure 3.2: The knee joint configurations are named by writing the knee joint orientation of hind legs/front legs together in sequence. As a result four types of knee config- uration are deduced: backward/backward, backward/forward, froward/back- ward and forward/forward. These four configurations are noted asBB,BF,FB,

FFfor short and shown from top-left to the bottom-right. Noting that footholds in all configurations are the same, however, location ofCoM(bigger black trian- gle) and its projection (smaller triangle) on horizontal plane are different.

3.2 movements of quadrupedal ditch crossing 29

3.1.3 Reachable Space of Foot on the Ground

The reachable space of foot on the ground, defined as the set of positions on the ground can be reached by the foot, has remarkable impact on the traversability and mobility of quadruped robots. The reachable space of foot are mainly determined by the motion range of joints in the leg, length of links of leg and the relative position between hip joint and ground or obstacle. In this section, to study the influence of morphological parameters including link length, we assume that the motion range of hip joint is α ∈ (−π, π) and the motion range of knee joint is β ∈ (−π, 0) or (0, π) for different knee orientation. This range is considerable larger than that of most of built legged robots, and will not constrain the movements of quadruped robot.

For two-link robotic leg, the reachable space of foot on the ground will also be limited by link length of leg and the distance from hip joint to ground in two distinct ways, shown in Fig. 3 .3. When ground A

D

is apart from hip joint further than the length of upper leg but still reachable for the foot of corresponding leg i.

e., l

6 h 6 l

+ l

, all locations in A

D

, shown in Fig. 3 .3a, are reachable for the foot, thus the reachable space of foot is A

D

. Whereas if distance h is smaller than upper leg length (h < l

), knee joint and upper leg will become interfered with ground AD, and then some part of the place in ground AD may be not reachable for foot, see Fig. 3 .3a. The lower leg at most poses horizontally and the section between BC are unreachable for foot.

In the other case, shown in Fig. 3 .3b, when h > l

but l

> h + l

, foot is not able to go through point O to the other side of hip joint and be only confined in one side of hip. Section BD in Fig. 3 .3b is reachable space.

Furthermore due to approaching singularity when legs pose near a straight line, the maximum length available l

for legs (black straight dash line HA or HD in Fig. 3 .3a) is constrained by a ratio c to be slightly shorter than the sum of upper and lower leg length. These three conditions are taken as constraints in simulation. And moreover we suppose the reachable space of foot like A

D

in Fig. 3 .3a is the nominal status for leg design and control, and other cases may lead to negative effect such as collision with ground like AD in Fig. 3 .3a or extra hip actuation demand like BD in Fig. 3 .3b. Consequently the constraint conditions can be written as:

max (l

, l

− l

) 6 h 6 l

(3)

l

= c (l

+ l

) . (4)

3.2 Movements of Quadrupedal Ditch Crossing

In order to develop a robot for natural environment deployment, features of ter-

rain where robots will be employed on should be analyzed at first. National In-

stitute of Standards and Technology (NIST) proposed a set of methods embodied

by American Society for Testing and Materials (

) to test the performance of

30 s t u d y o n t h e m o r p h o l o g i c a l pa r a m e t e r s c o n s i d e r i n g d i t c h c r o s s i n g t r av e r s a b i l i t y

(a)

(b)

Figure 3.3: Reachable space of foot on the ground. (a) demonstrates the effect when sup- port plane is nearer than the length of upper leg; (b) shows the case h > l1and l₂> h + l₁, in this situation foot is confined in one side of hip.

mobile robots including wheeled and tracked robot on challenging terrains such as ditch/gap [ASTM, 2011 a], hurdles [ASTM, 2011 b] and incline planes [ASTM, 2011 c]. We also refer these standards and select ditch as a benchmark to investigate the impacts of diverse morphological parameters, because of that the parameter used to describe a ditch in ground can be minimum, only ditch width W.

3.2.1 Ditch Crossing Procedure

While crossing a ditch, the movements of quadruped robot can be planned based on dynamic gait [Kalakrishnan et al., 2010 ] or static gait [Cheng and Pan, 1993 ].

In order to explore the potential limit contributed by morphological parameters independently, we take the static gait in simulation to rule out dynamic effect.

General movement sequences of quadruped robot coordinating its legs based on

static crawling gait is demonstrated in Fig. 3 .4. During this procedure, we assume

that the attitude of trunk are well regulated to maintain horizontal and the size of

foot tip is negligible thus two feet can locate at one place. In addition the longitude

stability margin is allowable to be zero at critical condition i.e.

is able to lie on

the boundary of support polygon. Morphological parameters adopted as demon-

stration in this simulation are slightly adjusted based on the structural data of

3.2 movements of quadrupedal ditch crossing 31

and HyQ2Max. The normalized morphological parameters of

, HyQ2Mx and demonstration are listed in Table 3 .2.

Table 3.2: Normalized Morphological Parameters of HyQ, HyQ2Max and Demonstration

Parameters HyQ HyQ2Max Demonstration

Length of trunk 1 1 1

Length of upper leg, ¯l

0 .47 0 .41 0 .45

Length of lower leg, ¯l

0 .48 0 .43 0 .45

Mass of trunk 1 1 1

Mass of upper leg, ¯ m

0 .06 0 .1 0 .1

Mass of lower leg, ¯ m

0 .02 0 .03 0 .06

Trunk

position, ¯r

(0, 0, 0) (0, 0, 0) (0, 0, 0)

Upper leg

position, ¯r

0 .47 0 .4 0 .4

Lower leg

position, ¯r

0 .34 0 .22 0 .5

Hip motion range, α −50 .. + 70 ±135 ±180

Knee motion range, β +20.. + 140 +2.. + 168 0.. + / − 180 Max. available leg length ratio, c 0 .94 0 .99 0 .9

Detailed strategy of ditch crossing is described as follows.

1 . Robot starts from standing posture with heading direction perpendicular to the edge of ditch (Fig. 3 .4(a)) and placing front feet at the starting edge of ditch.

2 . Robot moves its

towards the edge of ditch (Fig. 3 .4(b)) and stretches leg 3 backwards until its maximum length (Fig. 3 .4(c)) to move the

backwards, then place leg 4 at the starting edge of ditch as well (Fig. 3 .4(d)).

3 . Robot stretches leg 2 until maximum length to reach the other side (ending side) of the ditch, the

of robot will locate at the edge of starting side with the moving of leg 2 eventually because the robot here has a

configuration, which is symmetric geometrically with respect to the origin of robot frame when robot are in this posture (Fig. 3 .4(e)).

4 . Leg 3 is moved to the starting edge of ditch (Fig. 3 .4(f)), after that leg 1 stretches out to the ending side of ditch (Fig. 3 .4(g)).

5 . Robot moves

again forwards to the ending side of ditch (Fig. 3 .4(h)), then places the rest legs to the ending side of ditch in sequences as shown in Fig. 3 .4(i)-(l).

While crossing the ditch, the most important steps are critical postures when

the

of robot is just located above the edge of the ditch. Critical postures takes

place in two steps named as asses step and leaving step. Access step happens when

32 s t u d y o n t h e m o r p h o l o g i c a l pa r a m e t e r s c o n s i d e r i n g d i t c h c r o s s i n g t r av e r s a b i l i t y

the first front leg is up to touch the other side of the ditch, shown in Fig. 3 .4(e); and leaving step is when the last hind leg is up to leave touch with the starting side of the ditch, shown in Fig. 3 .4(j). The maximum width of ditch crossable eventually will be determined by one (or two) of these two steps. For some configurations asses step and leaving step may lead to distinct crossable ditch width. Fig. 3 .5 shows an asses step and a leaving step of

configuration as example to illustrate the width crossable. Ditch width W is the sum of two portions W = e

+ e

. e

is the protruding length of hip joints, which can be defined as the distance from protrud- ing hips to the edge of supporting side of ditch. e

is the horizontal projection of stretching leg which is independent from mass of limb and can be acquired from Eq. ( 5 ).

e

= q

l

− h

(5)

3.2.2 The Effect of Symmetry

According to the study above, it is notable that geometry symmetry exists in the knee configuration and the critical postures. For

and

configurations, the critical postures behaved in assess step and leaving step are the same exactly, see Fig.

3 .4(e) and (j), thus these two steps will lead to the same ditch width. In both assess step and leaving step,

projection is beneath of the origin of the robot frame due to geometric symmetry and then e

= 0.5, shown in Fig. 3 .6b.

For

and

configurations, the critical postures in assess step and leaving step are distinct, and then the ditch width crossable are different too. For

configura- tion, because links of legs tend to swing backwards, see Fig. 3 .6a, then the

of robot will move backwards and downwards with the increasing of link mass. This allows front hips to protrude more forwards then e

is increased in assess step but decreased in leaving step, shown in Fig. 3 .5. On the contrary for

configuration, e

will be decreased in assess step and increased in leaving step with the increasing of limb mass. Moreover, if other parameters except knee configuration are identical, the impacts on

and

configurations are exactly opposite. That means that the critical posture in assess step ( leaving step ) of

configuration is the same as that of

configuration in leaving step ( assess step), see Fig. 3 .5. Similar conclusions can be drawn as well for the effect of link’s

position for all four configurations.

3.3 Result and Discussion

The access step and leaving step of all four configurations are simulated in Matlab.

Fig. 3 .6a take the access step of

configuration as example to illustrate the proce-

dure of simulation. For a given combination of morphological parameters, the leg

2 and 3 are assumed to stretch outwards to their maximum lengths then compute

the footholds of leg 1 and 4 by iteration until the robot’s

projection on support

surface has the same longitude position (x coordinate) with the footholds of leg 1

and 4. Then we consider the position of footholds of leg 1 and 4 is the starting edge

3.3 result and discussion 33

of the ditch, and the distance between starting edge and the foothold of forward protruding leg are the width of ditch.

By varying the morphological parameters of quadruped robot, the influence on e1 and ditch width W is able to be concluded, and the results are shown in Fig. 3 .6 and Fig. 3 .7. For

and

configurations due to the geometrical symmetry, the impact on ditch width W and e

caused by mass of links ( ¯ m

, ¯ m

) and

position of links (¯r

, ¯r

) are zero, see surface B in Fig. 3 .6b and Fig. 3 .7a. The ditch width crossable is mainly determined by lengths of links (¯l

, ¯l

), shown by surface B in Fig. 3 .7b.

For

and

configuration, both mass related parameters and link length will lead to an obvious effect on the performance of ditch crossing. However the ef- fect on

and

configuration are opposite in the same critical step. Surface A in Fig. 3 .6b, Fig. 3 .7a and Fig. 3 .7b represent the influence of corresponding mor- phological parameters in access step of

configuration and the leaving step of

configuration. Similarly surface C in Fig. 3 .6b, Fig. 3 .7a and Fig. 3 .7b represent the influence of corresponding morphological parameters in leaving step of

config- uration and the access step of

configuration. Due to the ditch width W and e

represented by surface C is smaller than that represented by surface A, thus the values in surface C will be the ultimate ditch width W or e

crossable for

and

FF

configurations. Therefore, since the values of surface B are higher than that in surface C (shown in Fig. 3 .6b and Fig. 3 .6),

and

configurations are shown to be superior for ditch crossing.

Pitching of trunk can also affect the ditch crossing capability of robot by both changing mass distribution and varying the height of hips and then changing the reachable extent of feet. In actual cases, even though active control is applied, the torso may not always maintain horizontal attitude, its orientation may oscillate in a small range with respect to the desired zero pitch. To evaluate the influence of trunk tilting in small range and the height variation of

of trunk, a simulation is carried out on and shown in Fig. 3 .8. We assume trunk changes its pitch angle in the range of ±5 degrees and meanwhile the height of

of trunk increases form 0 to 0.4. Except varying parameters, the remained factors are kept the same as in Section 3 .2.1.

Based on the study above, the ditch crossing capability of

[Semini et al.,

2011 ] and HyQ2max [Semini et al., 2017 ] are analyzed. Normalized morpholog-

ical parameters can be seen in Table 3 .2. The limit of ditch crossing based on

kinematics and joint motion range is W = 1.053 (corresponding to 0.79m) for HyQ

and W = 1.217 (corresponding to 1.08m) for HyQ2Max. Due to limited hip motion

range of HyQ, although morphological parameters are similar to HyQ2max’s, the

hip height ( ¯h = 0.76) can not be lowered to near ground, then the stretch of leg

could not be utilized effectively comparing with HyQ2Max. It is notable that the

ditch crossing limit caused by morphological parameters are considerably large

comparing with the actual performance in experiments. Thus we can concluded

that the design including joint motion range and morphological parameters will

not constrain the ditch crossing capability, however, which may be confined by

other factors such as joint torque.

34 s t u d y o n t h e m o r p h o l o g i c a l pa r a m e t e r s c o n s i d e r i n g d i t c h c r o s s i n g t r av e r s a b i l i t y

Figure 3.4: Figures from (a) to (l) show the movement sequences of a quadruped robot with FB knee configuration crossing a ditch with maximum width. Parameters are selected according to the data of HyQ and HyQ2Max listed in Table3.2.

3.3 result and discussion 35

(a)

(b)

Figure 3.5: Critical postures ofBBduring ditch crossing. Due to all knee joints bend back- wards, the resultantCoMtend to be located backward with respect to the center of the trunk (origin of the robot frame), thus in (a) assess step, front hip joints can protrude more to increase the width of ditch. On the contrary, in (b) leav- ing step the protruding length of hind hip joints need to be reduced to prevent from tipping over backwards. Thus the ditch crossable in access step is larger than that in leaving step. As a result, forBB, the ultimate ditch width crossable is limited and determined by leaving step. For another situation that legs are too short to reach the beneath of the CoMeven though legs have extended to maximum length, then e1= e₂=p

l²_max− h²and W = 2p

l²_max− h². In this situation the ditch width crossable is merely associated with links’ length and relatively easy for analysis. Thus we mainly focus on the former case. Noting that (a) can also be considered as the leaving step ofFF configuration likewise (b) can be considered as the access step ofFFconfiguration

36 s t u d y o n t h e m o r p h o l o g i c a l pa r a m e t e r s c o n s i d e r i n g d i t c h c r o s s i n g t r av e r s a b i l i t y

(a)

(b)

Figure 3.6: Influence of link mass. (a) Taking BBconfiguration as an example to demon- strate simulation procedure. (b) Numerical result of the effect led by limb mass ( ¯m₁, ¯m₂) varying separately in the range of [0.1, 0.4] and [0.03, 0.23]. Surface A are the results ofBBconfiguration in access step andFFconfiguration in leaving step. Surface B are the results in access step and leaving step for bothFBandBF

configurations. Surface C are the results ofBBconfiguration in leaving step and

FFconfiguration in access step.

3.3 result and discussion 37

(a)

(b)

Figure 3.7: (a) Ditch width variation versus the change of linkage’sCoMposition. BothCoM

positions of upper leg ¯r1and lower leg ¯r2vary in the extent [0.1, 0.6]. (b) Ditch width W varies with the change of linkage length. Variation of ¯l1, ¯l2is in the range of [0.4, 0.8]. Surface A are the results of BB configuration in access step andFFconfiguration in leaving step. Surface B are the results in access step and leaving step for both FBand BFconfigurations. Surface C are the results ofBB

configuration in leaving step andFFconfiguration in access step.

38 s t u d y o n t h e m o r p h o l o g i c a l pa r a m e t e r s c o n s i d e r i n g d i t c h c r o s s i n g t r av e r s a b i l i t y

(a)

(b)

Figure 3.8: Figure (a) illustrates trunk’sCoMposition variation with a constant pitch angle θ = −5^◦. (b) Ditch width variation versus the pitch of trunk θ and the height ofCoMof trunk ¯r0z. Pitch angle of trunk θ changes from −5^◦to +5^◦ and the height of trunk’sCoM(¯r0z) varies in the range of [0, 0.4].

Structural Optimization of the Knee Joint 4

of HyQ2Max

HyQ2Max, presented for the first time in 2015 [Semini et al., 2015 ], is the latest generation of hydraulic quadruped robot developed by Dynamic Legged System (DLS) lab at Istituto Italiano di Tecnologia (IIT). The specifications of HyQ2Max and its predecessor —

have been presented in Table 2 .4. The main difference of mechanical system between

and HyQ2Max exists in the adopted actuators and the mechanical transmission of the joint. To realize more versatile movements, HyQ2Max has much more extensive workspace of foot and larger joint motion range than

’s. This chapter focuses on the knee joint of HyQ2Max and presents the actuation, transmission and torque profile optimization in details. Part of the content in this chapter has been published in [Semini et al., 2017 ].

4.1 Knee Joint Structure of HyQ2Max

The mechanical structure of the leg of HyQ2Max is introduced in section 2 .2.5. All of four legs have identical structure and are arranged at the corners of the trunk forming

configuration. The

and

joints of each leg are directly actuated by rotatory vane-type hydraulic motors, thus in the whole motion range, these two joints have constant torque limits that are independent from the joint angle. Due to the concern of inertia increasing, knee joint adopts a hydraulic cylinder as actuator instead of hydraulic motor that will leads to an improper mass distribution when being mounted at knee joint directly. In addition, a linkage mechanism connecting with the cylinder rod is designed to increase the joint motion range and generate desired knee joint torque profile. The detailed mechanical structure of knee joint is shown in Fig. 4 .1.

The relationship between actuating force exerted by hydraulic cylinder F

(p)

at pressure p and the output torque profile T (θ) with respect to knee joint can be expressed as:

T (θ) = J

(θ) · F

(p) . (6)

Where J

(θ) can be regarded as force/torque transmission ratio, generally, it is varying with respect to joint angle θ. F

(p) is proportional to the pressure of hydraulic fluid p. For the mechanism optimization in this chapter, we concern the extreme case in which the knee joint exerts maximum torque. So F

(p) can be considered as a constant as p = p

and denoted as F

for short. In order to obtain Eq. ( 6 ), two approaches are often used by researchers. The first approach

1 the force also depends on the direction for our unequal area cylinder

39

40 s t r u c t u r a l o p t i m i z at i o n o f t h e k n e e j o i n t o f h y q 2 m a x

Figure 4.1: Mechanical structure of the knee joint of HyQ2Max. The mechanism of the knee joint can be seen as the synthesis of a crank-slider mechanism consisting of the base AC, cylinder body 4 , cylinder rod 5 and link AR and a four- bar linkage including base AK, link AR, link BR and link KB connected with the lower leg rigidly. Considering the link AR and base are shared by the two mechanisms, therefore the combined mechanism includes 6 parts, is a six-bar mechanism. The other important components are: 1 servo valve from Moog, loadcell, 32 hydraulic manifold block, 4 5 hydraulic cylinder body and rod, absolute joint encoder with mounting bracket and 76 place for electronics.

αis the angle rotating from vector −→

AR to the direction of (positive) cylinder force−→

CR; β is the angle rotating from vector−→

AR to vector −→

BRand similarly, γis the angle rotating from vector−→

KBto vector−→

RB. Note that the coordinate for knee joint optimization in this chapter is defined independently from the coordinates defined for legs and the whole robot. The reason for this definition is the mechanism for all legs are the same and independent from the mounting orientation. The positive direction of related variables including forces and angles are defined based on the mechanism and indicated by arrows. When leg fully extends , the knee joint angle θ is defined as 0^◦. The motion range of KFEis from 2^◦ to 168^◦. Leg is nearly fully stretched at 2^◦ and the foot is contacted with the upper leg case mechanically at 168^◦. This figure is modified from [Semini et al.,2017].

4.1 knee joint structure of hyq2max 41

is based on the kinematics that is popular in the field of robotics and adopted in [Semini et al., 2017 ] as follows,

1 . deriving the geometric relationship between the length of hydraulic cylinder l

and the knee joint angle θ, acquiring the equation

l

= f(θ) ; (7)

2 . differentiating Eq. ( 7 ) to get the Jacobian J(θ), as ˙l

= J(θ) · ˙θ ; 3 . then joint torque profile T (θ) can be obtained as

T (θ) = J(θ) · F

(8)

4 . minimizing the difference C between T (θ) and desired torque profile T

. Due to complexity of the kinematics of closed loop mechanism and the differen- tiation of Eq. ( 7 ), the expression of Jacobian J(θ) will be rather complex and then the iteration computation of optimization will be very time-consuming.

The second approach can be derived based on mechanism analysis. Noting that the link AR is embraced by both crank-slider mechanism and four-bar linkage together, when the whole mechanism is in equilibrium, the resultant moment ap- plied on link AR by crank-slider mechanism and four-bar linkage should be zero.

The moment applied on a part caused by a force vector ~F can be computed by M = ~ ~r × ~F, where ~r is the displacement of force actuating point with respect to the pivot axis of part. In planar case, we use scalar form as M = F · r · sin ϕ, where ϕ is defined as the angle starting from ~r rotating to ~F, the sign or direction of mo- ment can be judged by analyzing part rotation in clock-wise or counter-clock-wise direction. Thus we have the following equations:

F

· l

· sinα = F

· l

· sinβ (9) T (θ) = F

· l

· sinγ. (10) Substituting Eq. 9 into Eq. 10 and eliminating l

, then joint output torque can be obtained as:

T (θ) = F

· l

·  sinα · sinγ sinβ



. (11)

Analyzing Eq. ( 11 ), it is notable that when angle α, β and γ equal 0 or π, the output torque T (θ) will be zero or infinite. Corresponding poses are singularity configurations of the mechanism. In practical design of knee joint, the cases of zero value of angle α, β and γ are eliminated intrinsically because of the interference between parts. As a result the singularities happens only when one or several of angles α, β and γ equal π. The different cases of singularities caused by single angle are analyzed below.

• When angle α equals π, hydraulic cylinder CR and link AR are inline. The

crank-slider mechanism including AC, cylinder body, cylinder rod and link

AR is in singularity configuration. The force exerted by hydraulic cylinder

is applied on base part directly through pivot A, therefore no actuation is

produced to move the joint, then T = 0.

42 s t r u c t u r a l o p t i m i z at i o n o f t h e k n e e j o i n t o f h y q 2 m a x

• When angle β equals π, link AR and BR are inline, the four-bar linkage com- prised by link AR, BR, KB and base AK is in singularity. In this position, the external torque applied to link KB and lower leg cannot induce rotation in link AR, even though the force of cylinder is zero. So the knee joint can af- ford infinite torque as long as the stress in mechanical structure is affordable.

This position is often called ’dead point’ in the field of mechanical design.

• when angle γ equals π, the link BR and KB are inline. This is another singu- larity configuration of the four-bar linkage. In this case, the effect is opposite to the above one: torque produced by link AR cannot lead to the rotation of link KB. And the force of hydraulic cylinder is transmitted to base part through pivot A and knee joint K.

According to the analysis above, we consider the singularity poses have negative influence on the transmission, so when we optimize the parameters, the singularity configurations ought to be avoided.

4.2 Objective Function and Constraints

In order to determine required torque profile for joints, 7 types of characteristic motions of HyQ2Max are taken into consideration [Semini et al., 2017 ] as func- tional metrics of the robot. The following list presents the 7 types of characteristic motions with desired performance levels.

1 . RT: walking trot on rough terrain [Barasuol et al., 2013 ] (0.5m/s) 2 . WT: walking trot on flat ground [Barasuol et al., 2013 ] (1.5m/s)

3 . TR: walking trot with turning [Barasuol et al., 2013 ] (0.5m/s with 25deg/s turning)

4 . PR: push recovery [Barasuol et al., 2013 ] (lateral perturbation of 500N for 1s) 5 . CF: crawling on flat ground [Winkler et al., 2014 ] (average speed 0.1m/s) 6 . CS: stair climbing [Winkler et al., 2015 ] (step height 0.12m and step depth

0 .3m)

7 . SR: self-righting (predefined motion as described in [Semini et al., 2015 ])

Combining the characteristic motions with morphological parameters of HyQ2Max

including limb lengths, masses of parts, etc, a plenty of dynamic simulations for

characteristic motions are performed in an simulation environment developed by

Dynamic Legged System (

) lab based on Simulation Laboratory (

)[Schaal,

2009 ] and Robotics Code Generator (

) [Frigerio et al., 2016 ]. Moreover,

in some simulations, the payload afforded by HyQ2Max is applied by means of

assigning different additional mass to the trunk in software. Finally the result of

simulations of characteristic motions with 40 kg extra mass are adopted as the

4.2 objective function and constraints 43

object for robot design. Fig. 4 .2a presents the exerted torques vs. joint angle plots of

leg, the direction and sign of torque and joint angle follow the conventional definition of

in [Semini, 2010 ]. Fig. 4 .2b shows the exerted torques with re- spect to joint angular position of all four legs, and the desired functions of torque profile for sizing actuators are also indicated by green dash lines. For

joints, the torques can be both positive and negative and the positive torques is obvi- ously larger than negative torques in magnitude, approximately 5 : 1. Since the knee joints of legged robot carry the weight of the robot so the torque is mostly to extend the leg not retract. So when sizing the diameter of hydraulic cylinder, we use the side of piston without cylinder rod to produce positive torque and the other side of piston with cylinder rod to generate negative torque. According to our survey on the small-diameter (piston diameter 6 40 mm) hydraulic cylinders from several different companies including SMC [SMC Company, 2001 ], Hoer- biger [Hoerbiger Company, 2007 ] and Bansbach [Bansbach Company, 2012 ], the ratio of rod diameter and piston diameter is nearly 1 : 2. Then the force generated by the two sides of the piston at the same pressure is about 3 : 4. Thus the selection of the diameter of the hydraulic cylinder should be done according to the positive torque profile (dash line 5 in Fig. 4 .2b), and then the negative torque profile (dash line 6 in Fig. 4 .2b) can be covered by itself.

A polynomial function for the dash line 5 in Fig. 4 .2b is constructed by fitting selected points with proper coordinates in MATLAB. The desired torque function can be written as:

T

(θ) = 3 .844 × 10

θ

− 1 .137 × 10

θ

+ 9 .186 × 10

θ

+ 7.096 × 10

θ

+ 1.154 θ + 133. (12) And the cost function to be minimized is defined as the maximum absolute value of the difference between Eq. ( 11 ) and Eq. ( 12 ), written as:

C = max |T(θ) − T

(θ) | . (13)

Where θ is the

joint angle ranging from 2

to 168

. Practically we are only interested in the range from 40

to 168

in optimization because in the range 2

to 40

, no torque exerts by knee joints in all the simulations for characteristic motions.

According to the structure presented in Fig. 4 .1, we select the coordinates of

pivot A and C, the length of link AR, BR, KB, ∠FKB (the angle between lower

leg KF and link KB) and the ares of hydraulic cylinder piston S

as variables

for optimization. The initial values and the upper/lower boundaries of each vari-

able are listed in Table 4 .1. Moreover the variables should subject to the following

44 s t r u c t u r a l o p t i m i z at i o n o f t h e k n e e j o i n t o f h y q 2 m a x

Joi nt

Joint position [Degrees]

(a)

Joi nt

Joint position [Degrees]

(b)

Figure 4.2: Joint torques of 7 characteristic motions. (a) Joint torques ofLFleg of diverse characteristic motions with 40 kg payload; (b) joint torques vs. joint angle plots of 4 legs are merged together. The signs of torque and joint angle are accord- ing to the convention in [Semini,2010]. Green dash lines indicate the desired function of torques with respect to joint angles. ForHAAandHFEactuated by hydraulic motors, the desired functions of torque with respect to joint angle can be straight line, because at every angular position the motors can output the same torque. ForKFEactuated by a hydraulic cylinder, a polynomial func- tion is constructed as desired torque function, which can cover the joint torque requirements of characteristic motions.

4.3 optimization result 45

Table 4.1: Optimization Variables

Description Variable Initial value Upper boundary Lower boundary Unit

Pivot A A

320 360 280 mm

A

-23 10 -50.5 mm

Pivot C C

-41 -13 -50 mm

C

5 47 -47 mm

Link AR l

50 70 25 mm

Link BR l

50 70 25 mm

Link KB l

34 34 .5 32 mm

Angular offset ∠FBK 39 .5 39 40

Piston area S

310 250 450 mm

inequality constraints according to the available space inside the upper leg shell:

37 6 q

C

+ C

6 48 (14)

30 6 q

(A

− K

)

+ (A

− K

)

(15) q

(A

− x

)

+ (A

+ y

)

> 62 (16) q

(A

− x

)

+ (A

− y

)

> 64 + l

(17) 13

280 A

− A

− 53 6 0 (18)

Where the inequality constraints limit the positions of pivot A and C within the pink regions illustrated in Fig. 4 .1, and (x

, y

) = (309.7, 100.7). The output ob- tained from the optimizer is: A

= 338 .6mm, A

= −30 .3mm, C

= −41 .4mm, C

= 10 .2mm, l

= 70 .0mm, l

= 59 .4mm, l

= 32 .0mm, S

= 418mm

and

∠FBK = 39

.

4.3 Optimization Result

To verify the result of optimization based on Matlab, a simulation of 6-bar mecha- nism is performed in Automated Dynamic Analysis of Mechanical Systems (

).

ADAMS

is one of the world’s most famous and widely used Multibody Dynamics

(

) software, which has trustable performance in the field of mechanism simu-

lation, paralytically when mechanism is complex and include closed loop. A phys-

ical model of the 6-bar mechanism is built in

based on the optimization

result above. In the model a force equivalent to the force produced by hydraulic

cylinder is applied and a rotational motion is added onto the lower leg to drive

the whole mechanism. Then for the rotational motion, the torque demanded to

overcome the force will be the torque generated by the actuator but with opposite

sign. The model of mechanism and the result plots are presented in Fig. 4 .3. Com-

46 s t r u c t u r a l o p t i m i z at i o n o f t h e k n e e j o i n t o f h y q 2 m a x

paring with the data outputted from

and Matlab, the curves are matching.

Moreover the maximum error of optimization and the values of angle α, β and γ

with respect to θ during motion are also presented in Fig. 4 .4.

4.3 optimization result 47

Figure 4.3: Knee joint simulation in ADAMS. (Top) The motion sequences of 6-bar mech- anism simulation in ADAMS. (Bottom) Simulation results: (red) torque profile optimized, (blue) length of hydraulic cylinder.